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Abstract

We have used a 3-GHz microwave host cavity to study the remarkable electronic properties
of metallic, single-walled carbon nanotubes. Powder samples are placed in its magnetic
field antinode, which induces microwave currents without the need for electrical contacts.
Samples are shown to screen effectively the microwave magnetic field, implying an
extremely low value of sheet resistance (< 10 μΩ) within the graphene sheets making
up the curved nanotube walls. Associated microwave losses are large due to the large
surface area, and also point to a similar, very small value of sheet resistance due
to the inherent ballistic electron transport.

Keywords:

73.63.Fg; 72.80.Rj; 78.70.Gq; Carbon nanotubes; Microwave; Screening

Background

There has been intensive research on the characterisation and application of carbon
nanotubes for next-generation electronic devices [1]. Ballistic conduction along single-walled metallic nanotubes is predicted theoretically
and has been experimentally verified. Clearly, this has enormous impact for electronic
devices fabricated using these remarkable materials, but there are problems in characterising
the electronic properties using conventional (i.e. DC) techniques. This is mainly
due to the need to make electrical contacts with the ends of the nanotube, which suppresses
the measured conductivity, and is further complicated by inter-tube contacts. For
example, it has been found that thin film samples have sheet resistances limited by
the large junction resistances between adjacent nanotubes [2].

In this context, we propose a new method to characterise the electronic properties
of metallic nanotubes using microwaves as a means of exciting currents, thus negating
the requirement for electrical contacts. The method is fast and is performed on a
powder sample, so it allows rapid measurements of large numbers of samples and permits
in situ changes in electronic properties (e.g. due to surface adsorption of chemical species)
without the need to remove the sample from the apparatus. Previous microwave studies
of carbon nanotube materials have ranged from the screening efficiency of thin films
[3] to the microwave excitation of single nanotubes [4]. In our experiments, powder samples are placed in a region of high microwave magnetic
field in a microwave cavity (here, a 3-GHz copper hairpin), and the sample properties
are determined using the cavity perturbation technique. We have used a similar method
to measure the electronic properties of a range of other material systems (e.g. metal-loaded
zeolites [5]).

Methods

Powdered nanotube samples were obtained from Thomas Swan & Co. Ltd. (County Durham,
UK), prepared by thermal CVD at 800°C, giving a mix of semiconducting and metallic
single-walled nanotubes. Figure 1 shows the scanning electron microscope (SEM) image of the nanotubes before and after
manual grinding. The long nanotubes are broken into smaller segments confirming the
production of short carbon nanotubes, which has been previously reported using the
ball milling technique.

Figure 1.SEM image of the nanotubes used in this experiment. (a) Before and (b,c) after manual grinding.

Analysis of the transmission electron microscope (TEM) images shown in Figure 2 suggests the nanotubes have diameters ranging from 1.2 to 1.7 nm and lengths of 100 nm
to 4 μm. Scanning transmission microscopy (STEM) of the nanotubes shows that there
are iron (Fe) impurities on the surface of the nanotubes. Energy dispersive X-ray
spectroscopy gives atomic percent ratios of 96.5:2.7:0.8 for C:O:Fe, respectively.
Samples were loaded into low-loss quartz tubes (inner diameter of 1.3 mm and wall
thickness 0.35 mm) and placed parallel to the microwave magnetic field in the 3-GHz
quarter-wave copper hairpin cavity, shown in Figure 3.

The hairpin itself is a 1-mm thick copper strip, with a width of 5 mm, bent into a
U shape of length 23 mm and plate separation of 5 mm. The expected resonant frequency
of the fundamental mode of the hairpin is, therefore, f0 ≈ c/4 L ≈ 3.26 GHz, in practise about 10% lower due to the end effects of electric and magnetic
fields spilling outside of the structure. The microwave magnetic field is largest
at the short-circuit end of the hairpin, where the field is also approximately uniform
and runs parallel to the plate surfaces. The sample is inserted into this region.
The hairpin is surrounded by a cylindrical radiation shield, of length twice that
of the hairpin (i.e. 50 mm) and inner radius of 15 mm. This reduces the radiation
losses from the hairpin and, thus, preserves a high quality factor, which is desirable
for measuring the additional microwave loss contributed by the sample. Microwave power
is coupled in and out of the hairpin using an identical pair of loop-terminated RG405
microwave coaxial cables which couple to the microwave magnetic field at the hairpin’s
short-circuit end. The transmitted microwave power (i.e. |S21|2) is measured in the frequency domain using an Agilent E5071B network analyser with
computer control (Agilent Technologies, Inc., CA, USA). The unloaded quality factor
Q of the empty cavity is around 1,500, i.e. large for its small volume (≈ 600 mm3), thus enhancing the sample filling factor for more sensitive measurements of its
electronic properties. Cavity couplings are kept weak (i.e. insertion loss > 30 dB
at resonance), so that only minor correction factors are needed to extract unloaded
Q factors from the raw data.

Results and discussion

Microwave results for a nanotube sample are shown in Figure 4, compared with a similar volume of graphite powder. Remarkably, this nanotube sample
exhibits screening of the microwave magnetic field. This is evidenced by an increased
resonant frequency of Δf0 = 9.0 ± 0.3 MHz relative to the empty tube. There are also very large microwave losses,
evidenced by an increased 3-dB bandwidth of ΔfB = 18 ± 1 MHz. Such screening is usually observed only in powdered (or bulk) metals
and of conductivity large enough for the mean particle size to become much greater
than the microwave skin depth. As can be seen from the data of Figure 4, the conductivity of graphite is not high enough to observe any such screening; rather,
it exhibits the response typical of a lossy dielectric (i.e. a broadened 3-dB bandwidth
with no frequency shift, indicative of full sample penetration by the microwave magnetic
field).

Figure 4.Resonant traces for the nanotube sample compared with graphite power. The increased resonant frequency of the nanotube sample is due to effective screening
of the

H

-

field.

To analyse this data, we use standard cavity perturbation theory, which assumes a
small sample filling fraction within the cavity [6]. For the

H

-field applied parallel to the nanotube axis (Figure 5a), the main results are the following:

(1)

(2)

where Vs is the total sample volume; Veff, effective cavity volume; and τ, relaxation time, defined via

(3)

with a being the average nanotube radius, H0 the local magnetic field strength at the sample and Rsq the sheet resistance of graphene walls of the nanotubes. As can be seen from Figure 5, both axial and radial components of

H

relative to the nanotube axis will be screened if the sheet resistance is low enough.
Effective screening only occurs in the limit when ωτ > 1, i.e. when Rsq < μ0aω/2.

Figure 5.Screening currents for the microwave-field. Applied (a) parallel and (b) perpendicular to the nanotube axis.

A separate calibration experiment involved the insertion of small copper spheres of
radii 330 μm into the same

Conclusions

The inner tube volume within the cavity is ≈ 6.6 mm3, of which ≈ 0.40 mm3 is nanotube material (i.e. a packing fraction of about 6% by volume, based on the
sample mass of 1.3 mg). From this, we conclude that the magnetic field is effectively
screened within this low density powder, both from within the nanotubes themselves
and from a much larger volume (×3) in the space outside. For this to happen, we also
conclude that the nanotube sheet resistance must be extremely small, i.e. an upper
limit of Rsq ≈ 10 μΩ based on a mean nanotube radius of a ≈ 0.7 nm. Since a trace of metallic Fe is present in the sample, it is prudent to
quantify how much this contributes to the microwave screening. The 0.8% atomic ratio
of Fe atoms corresponds to a volume of about 0.006 mm3. This is likely to be dispersed as very small particles of radii less than 1 μm.
Given that the skin depth of Fe at 3 GHz at room temperature is about 2.8 μm, we would
not expect such small particles to screen the microwave magnetic field. Even if the
Fe formed a single large particle, which gives rise to the greatest screened volume,
the contribution to increase in resonant frequency will be only 0.6% of that observed
for the whole CNT sample. Hence, we conclude that the screening effect of the Fe is
negligible.

Does such a small value of Rsq contradict the observation of the large microwave losses of Figure 4? The answer can be found in the huge surface area of carbon nanotube powders, e.g.
our 1.3-mg sample has a total surface area S ≈ 1.1 m3. Sheet resistance can be extracted from Equation 1b in the limit of strong screening
ωτ > > 1, resulting in Rsq ≈ 2πμ0Veff ΔfB / S ≈ 30 μΩ. Therefore, these two independent measurements (one of microwave screening,
the other of microwave loss) both point to very small values of sheet resistance due
to ballistic transport. In experimental studies of carbon nanotube films, the sheet
resistance is found to be of the order of 1 kΩ [7] with similar results for graphene films [8]. In these studies, the intrinsic sheet resistance of the carbon nanotubes can be
difficult to extract from the measurements as charge transport is limited by contacts
between the nanotubes. Other types of measurement such as the optical conductivities
reported in [8] do not circumvent this problem as they characterise relatively large areas of the
films. In theoretical studies of devices based on discrete carbon nanotubes and of
carbon nanotube films, charge transport is often assumed to be ballistic, and therefore,
the sheet resistance is zero. Ballistic transport has been observed in ohmically contacted
metallic single wall carbon nanotubes having lengths less than approximately 300 nm
[9].

Not all of the nanotube powders studied to date show such striking behaviour, so future
experiments will concentrate on systematic studies of a wider range of materials (semiconducting
and metallic), over a wider range of frequencies (particularly from kilohertz to megahertz),
also trying to identify the nature of the defects giving rise to the finite sheet
resistance. Indeed, isolating metallic samples is, itself, an important problem, and
the method we have proposed here may serve as a means of quantifying the volume fraction
of metallic nanotubes within a given powder.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

AP and DIO carried out the microwave measurements, and DIO characterized the sample.
AP and PAC conceived of the study and completed the analysis. All authors read and
approved the final manuscript.